Relativistic virialization in the Spherical Collapse model for Einstein-de Sitter and \Lambda CDM cosmologies

TL;DR

This paper develops a fully relativistic virial theorem for spherical collapse in cosmology, comparing it with classical approaches and quantifying relativistic corrections in Einstein-de Sitter and mbda CDM models.

Contribution

It introduces a general relativistic virial theorem based on TOV solutions, providing a more fundamental perspective on virialization in cosmological structure formation.

Findings

Relativistic corrections are of order 10^{-3}% in the weak field limit.

The relativistic approach reproduces the classical solution of Wang & Steinhardt (1998).

The method offers potential insights into the virialization process beyond classical mechanics.

Abstract

Spherical collapse has turned out to be a successful semi-analytic model to study structure formation in different DE models and theories of gravity. Nevertheless, the process of virialization is commonly studied on the basis of the virial theorem of classical mechanics. In the present paper, a fully generally-relativistic virial theorem based on the Tolman-Oppenheimer-Volkoff (TOV) solution for homogeneous, perfect-fluid spheres is constructed for the Einstein-de Sitter and \Lambda CDM cosmologies. We investigate the accuracy of classical virialization studies on cosmological scales and consider virialization from a more fundamental point of view. Throughout, we remain within general relativity and the class of FLRW models. The virialization equation is set up and solved numerically for the virial radius, y_{vir}, from which the virial overdensity \Delta_{V} is directly obtained. Leading order corrections in the post-Newtonian framework are derived and quantified. In addition, problems in the application of this formalism to dynamical DE models are pointed out and discussed explicitly. We show that, in the weak field limit, the relative contribution of the leading order terms of the post-Newtonian expansion are of the order of 10^{-3}% and the solution of Wang & Steinhardt (1998) is precisely reproduced. Apart from the small corrections, the method could provide insight into the process of virialization from a more fundamental point of view.